The Kinetic Hourglass Data Structure for Computing the Bottleneck Distance of Dynamic Data

TL;DR

The paper presents the kinetic hourglass, a novel kinetic data structure for maintaining the bottleneck distance of a weighted bipartite graph under time-varying edge weights, built as two interlocked priority structures that track the current bottleneck edge via a shared root. Grounded in the kinetic data structure framework, it analyzes certificates, internal/external events, and updates, offering two implementation paths: the deterministic kinetic heap and the randomized kinetic hanger. As a primary application, the authors demonstrate an exact computation of the bottleneck distance between two persistent homology transforms (PHTs) in $\mathbb{R}^2$, focusing on star-shaped objects to control monodromy, and deriving concrete complexity bounds (up to $O(n^6\log n)$ when the edge set scales as $4n^2$). The work provides a pathway for exact vine-based comparisons of PHTs and related vineyards, with potential extensions to XPHT and Wasserstein-type distances, contributing a new exact, dynamical perspective to topological shape analysis. The framework blends kinetic heap/hanger techniques with persistent homology, enabling precise, dynamic comparisons that were previously approached only via sampling or static reductions.

Abstract

The kinetic data structure (KDS) framework is a powerful tool for maintaining various geometric configurations of continuously moving objects. In this work, we introduce the kinetic hourglass, a novel KDS implementation designed to compute the bottleneck distance for geometric matching problems. We detail the events and updates required for handling general graphs, accompanied by a complexity analysis. Furthermore, we demonstrate the utility of the kinetic hourglass by applying it to compute the bottleneck distance between two persistent homology transforms (PHTs) derived from shapes in $\mathbb{R}^2$, which are topological summaries obtained by computing persistent homology from every direction in $\mathbb{S}^1$.

Figures (5)

• Figure 1: Construction of the bipartite graph $G$ based on the persistence diagrams $X$ and $Y$.
• Figure 2: Kinetic heap event updates
• Figure 3: Illustration of construction of the kinetic hourglass.
• Figure 4: Illustration of Scenario 1 of and $M$-Event; see Lemma \ref{['lem:M']}.
• Figure 6: An example of the weight function $\text{c}$. The two figures in the middle visualize the behavior of the vines. On the top right is the bipartite graph representation, and on the bottom are the cost functions.

Theorems & Definitions (20)

• Theorem 2.1: Hall's Marriage Theorem
• Definition 2.2
• Definition 2.3
• Theorem 2.4: Berge's Theorem
• Definition 2.5
• Lemma 2.6: Reduction Lemma EdelsHarer2010
• Lemma 3.1
• proof
• Lemma 3.2: $L$-Event
• proof